A024207 Number of terms in n-th derivative of a function composed with itself 7 times.

1, 1, 7, 28, 105, 322, 952, 2541, 6539, 15833, 37148, 83594, 183289, 389520, 809820, 1643375, 3272797, 6390745, 12279337, 23208483, 43252360, 79483096, 144265338, 258673983, 458747540, 804877837, 1398356706, 2406328974, 4104352128, 6940717598, 11643270856

Offset: 0

Author

Winston C. Yang (yang(AT)math.wisc.edu)

Keywords

• nonn

References

• W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Crossrefs

Cf. A008778, A022811-A022817, A024208-A024210. First column of A050301.

Column k=7 of A022818.

Programs

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n < k, 0, If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k - j], {j, 0, Min[n/i, k]}]]]];
a[n_, k_] := a[n, k] = If[k == 1, 1, Sum[b[n, n, i]*a[i, k-1], {i, 0, n}]];
a[n_] := a[n, 7];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 28 2017, after Alois P. Heinz *)

Formula

If a(n,m) = number of terms in m-derivative of a function composed with itself n times, p(n,k) = number of partitions of n into k parts, then a(n,m) = sum_{i=0..m} p(m,i)*a(n-1,i).

Extensions

More terms from Alois P. Heinz, Aug 18 2012